\(\int \frac {x^3}{(1-x^3) (1+x^3)^2} \, dx\) [450]

Optimal result

Integrand size = 20, antiderivative size = 97 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=-\frac {x}{6 \left (1+x^3\right )}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log (1-x)-\frac {1}{36} \log (1+x)+\frac {1}{72} \log \left (1-x+x^2\right )+\frac {1}{24} \log \left (1+x+x^2\right ) \] Output:

-1/6*x/(x^3+1)+1/36*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/12*arctan(1/3*(1 
+2*x)*3^(1/2))*3^(1/2)-1/12*ln(1-x)-1/36*ln(1+x)+1/72*ln(x^2-x+1)+1/24*ln( 
x^2+x+1)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=\frac {1}{72} \left (-\frac {12 x}{1+x^3}-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+6 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-6 \log (1-x)-2 \log (1+x)+\log \left (1-x+x^2\right )+3 \log \left (1+x+x^2\right )\right ) \] Input:

Integrate[x^3/((1 - x^3)*(1 + x^3)^2),x]
 

Output:

((-12*x)/(1 + x^3) - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 6*Sqrt[3]*ArcT 
an[(1 + 2*x)/Sqrt[3]] - 6*Log[1 - x] - 2*Log[1 + x] + Log[1 - x + x^2] + 3 
*Log[1 + x + x^2])/72
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {971, 1020, 750, 16, 1142, 25, 1083, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3}{\left (1-x^3\right ) \left (x^3+1\right )^2} \, dx\)

\(\Big \downarrow \) 971

\(\displaystyle \frac {1}{6} \int \frac {2 x^3+1}{\left (1-x^3\right ) \left (x^3+1\right )}dx-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 1020

\(\displaystyle \frac {1}{6} \left (\frac {3}{2} \int \frac {1}{1-x^3}dx-\frac {1}{2} \int \frac {1}{x^3+1}dx\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (-\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx-\frac {1}{3} \int \frac {1}{x+1}dx\right )+\frac {3}{2} \left (\frac {1}{3} \int \frac {x+2}{x^2+x+1}dx+\frac {1}{3} \int \frac {1}{1-x}dx\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (-\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \int \frac {x+2}{x^2+x+1}dx-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2+x+1}dx+\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2+x+1}dx+\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx\right )-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{3} \left (3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx-3 \int \frac {1}{-(2 x+1)^2-3}d(2 x+1)\right )-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{3} \left (-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {2 x+1}{x^2+x+1}dx+\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \log \left (x^2-x+1\right )-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^2+x+1\right )\right )-\frac {1}{3} \log (1-x)\right )\right )-\frac {x}{6 \left (x^3+1\right )}\)

Input:

Int[x^3/((1 - x^3)*(1 + x^3)^2),x]
 

Output:

-1/6*x/(1 + x^3) + ((-1/3*Log[1 + x] + (-(Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3 
]]) + Log[1 - x + x^2]/2)/3)/2 + (3*(-1/3*Log[1 - x] + (Sqrt[3]*ArcTan[(1 
+ 2*x)/Sqrt[3]] + Log[1 + x + x^2]/2)/3))/2)/6
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*Rt[a, 3] - 
 Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; 
 FreeQ[{a, b}, x]
 

rule 971

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* 
((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) 
*(p + 1))   Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - 
 n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e 
, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 
 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {x}{6 \left (x^{3}+1\right )}+\frac {\ln \left (4 x^{2}-4 x +4\right )}{72}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{36}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{24}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {\ln \left (x -1\right )}{12}-\frac {\ln \left (x +1\right )}{36}\)	\(82\)
default	\(\frac {-2 x -2}{36 x^{2}-36 x +36}+\frac {\ln \left (x^{2}-x +1\right )}{72}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{36}+\frac {\ln \left (x^{2}+x +1\right )}{24}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}+\frac {1}{18 x +18}-\frac {\ln \left (x +1\right )}{36}-\frac {\ln \left (x -1\right )}{12}\)	\(90\)

Input:

int(x^3/(-x^3+1)/(x^3+1)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/6*x/(x^3+1)+1/72*ln(4*x^2-4*x+4)-1/36*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2 
))+1/24*ln(4*x^2+4*x+4)+1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/12*ln(x 
-1)-1/36*ln(x+1)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=\frac {6 \, \sqrt {3} {\left (x^{3} + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left (x^{3} + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 3 \, {\left (x^{3} + 1\right )} \log \left (x^{2} + x + 1\right ) + {\left (x^{3} + 1\right )} \log \left (x^{2} - x + 1\right ) - 2 \, {\left (x^{3} + 1\right )} \log \left (x + 1\right ) - 6 \, {\left (x^{3} + 1\right )} \log \left (x - 1\right ) - 12 \, x}{72 \, {\left (x^{3} + 1\right )}} \] Input:

integrate(x^3/(-x^3+1)/(x^3+1)^2,x, algorithm="fricas")
 

Output:

1/72*(6*sqrt(3)*(x^3 + 1)*arctan(1/3*sqrt(3)*(2*x + 1)) - 2*sqrt(3)*(x^3 + 
 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + 3*(x^3 + 1)*log(x^2 + x + 1) + (x^3 + 
1)*log(x^2 - x + 1) - 2*(x^3 + 1)*log(x + 1) - 6*(x^3 + 1)*log(x - 1) - 12 
*x)/(x^3 + 1)
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=- \frac {x}{6 x^{3} + 6} - \frac {\log {\left (x - 1 \right )}}{12} - \frac {\log {\left (x + 1 \right )}}{36} + \frac {\log {\left (x^{2} - x + 1 \right )}}{72} + \frac {\log {\left (x^{2} + x + 1 \right )}}{24} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{36} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \] Input:

integrate(x**3/(-x**3+1)/(x**3+1)**2,x)
 

Output:

-x/(6*x**3 + 6) - log(x - 1)/12 - log(x + 1)/36 + log(x**2 - x + 1)/72 + l 
og(x**2 + x + 1)/24 - sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/36 + sqrt(3) 
*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/12
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {x}{6 \, {\left (x^{3} + 1\right )}} + \frac {1}{24} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{72} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{36} \, \log \left (x + 1\right ) - \frac {1}{12} \, \log \left (x - 1\right ) \] Input:

integrate(x^3/(-x^3+1)/(x^3+1)^2,x, algorithm="maxima")
 

Output:

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/36*sqrt(3)*arctan(1/3*sqrt( 
3)*(2*x - 1)) - 1/6*x/(x^3 + 1) + 1/24*log(x^2 + x + 1) + 1/72*log(x^2 - x 
 + 1) - 1/36*log(x + 1) - 1/12*log(x - 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{36} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {x}{6 \, {\left (x^{3} + 1\right )}} + \frac {1}{24} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{72} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{36} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{12} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:

integrate(x^3/(-x^3+1)/(x^3+1)^2,x, algorithm="giac")
 

Output:

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/36*sqrt(3)*arctan(1/3*sqrt( 
3)*(2*x - 1)) - 1/6*x/(x^3 + 1) + 1/24*log(x^2 + x + 1) + 1/72*log(x^2 - x 
 + 1) - 1/36*log(abs(x + 1)) - 1/12*log(abs(x - 1))
 

Mupad [B] (verification not implemented)

Time = 3.73 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=-\frac {\ln \left (x-1\right )}{12}-\frac {\ln \left (x+1\right )}{36}-\frac {x}{6\,\left (x^3+1\right )}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{72}+\frac {\sqrt {3}\,1{}\mathrm {i}}{72}\right ) \] Input:

int(-x^3/((x^3 - 1)*(x^3 + 1)^2),x)
 

Output:

log(x + (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*1i)/24 + 1/24) - log(x + 1)/36 - x 
/(6*(x^3 + 1)) - log(x - (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*1i)/24 - 1/24) - 
log(x - 1)/12 + log(x - (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/72 + 1/72) - l 
og(x + (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/72 - 1/72)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.58 \[ \int \frac {x^3}{\left (1-x^3\right ) \left (1+x^3\right )^2} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) x^{3}-2 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right )+6 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{3}+6 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )+\mathrm {log}\left (x^{2}-x +1\right ) x^{3}+\mathrm {log}\left (x^{2}-x +1\right )+3 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{3}+3 \,\mathrm {log}\left (x^{2}+x +1\right )-6 \,\mathrm {log}\left (x -1\right ) x^{3}-6 \,\mathrm {log}\left (x -1\right )-2 \,\mathrm {log}\left (x +1\right ) x^{3}-2 \,\mathrm {log}\left (x +1\right )-12 x}{72 x^{3}+72} \] Input:

int(x^3/(-x^3+1)/(x^3+1)^2,x)
 

Output:

( - 2*sqrt(3)*atan((2*x - 1)/sqrt(3))*x**3 - 2*sqrt(3)*atan((2*x - 1)/sqrt 
(3)) + 6*sqrt(3)*atan((2*x + 1)/sqrt(3))*x**3 + 6*sqrt(3)*atan((2*x + 1)/s 
qrt(3)) + log(x**2 - x + 1)*x**3 + log(x**2 - x + 1) + 3*log(x**2 + x + 1) 
*x**3 + 3*log(x**2 + x + 1) - 6*log(x - 1)*x**3 - 6*log(x - 1) - 2*log(x + 
 1)*x**3 - 2*log(x + 1) - 12*x)/(72*(x**3 + 1))